Abstract—In soft object modeling, primitive soft objects can

be used to construct a complex soft object by performing

addition operations only. This is because a primitive soft object

is defined as the iso-surface of a field function which is a

composition of a potential function and a distance function. In

fact, the distance function determines the shape of a soft object.

To deform the shape of a primitive soft object, this paper

proposes dilation and erosion operations, which can be viewed

as distance function operations because they perform on the

distance function of a primitive soft object to be deformed and a

chosen dilating or eroding distance function. Thus, dilation

operation can be used to dilate a primitive soft object through

the chosen dilating distance function, and erosion operation can

be used to erode a primitive soft object through the chosen

eroding distance function. Briefly, these two operations can

deform the iso-surface of an existing distance function and then

can be applied as a new distance function to define a new field

function.

Index Terms—Field functions, distance functions, soft

objects.

I. INTRODUCTION

In implicit surfaces, primitive implicit surfaces are defined

each as an iso-surface of a defining function. A complex

implicit surface is created by smoothly connecting primitive

surfaces via blending operations [1], [2]. Among different

representations of implicit surfaces [3], [4], [5], soft object

modeling especially can blend primitive soft objects easily by

soft blending [4], [5] which performs addition operations

only. This is because field functions are used as defining

functions and the value of a field function is designed to be

decreasing from 1 to 0. More precisely, a field function is

defined as a composition of a potential function and a

distance function. Potential functions majorly ensure a field

function to be decreasing and controls the softness affect of

soft blending such as those in [4], [5], [6], [7], and the one in

[7] particularly offers blending range control. Distance

function controls the shape of a primitive soft object.

Existing distance functions include spheres [5],

super-ellipsoids [6], [8], super-quadrics [3], generalized

distance functions [9], skeletal primitives [10], and sweep

objects [11].

Regarding existing distance functions, most of the

researches focus directly on developing new distance

functions with new shapes of iso-surfaces. However,

different from their developing methods, this paper proposes

a new method to develop a new distance function by

Manuscript received August 20, 2012; revised September 15, 2012.

Pi-Chung Hsu is with the Department. of Information Management,

Shu-Te University, Kaohsiung, Taiwan (e-mail: pichung@stu.edu.tw).

deforming the iso-surface of a distance function via another

distance function via newly developed distance function

operations on the above two distance functions. That is, this

paper develops distance function operations and then existing

distance functions are applied into them to create a new

distance function for generating a new field function with

deformed iso-surfaces. These distance function operations

are described as follows:

Erosion operation to erode the distance function of a soft

object by an eroding distance function:

When applied as a new distance function to deform a soft

object, it can be used to shrink a soft object with different

magnitudes for different directions determined by the

eroding distance function, and it also offers a parameter to

control the degree to which a soft object can be shrunk.

Dilation operation to dilate the distance function of a soft

object by a dilating distance function:

When applied as a new distance function to deform a soft

object, it can be used to enlarge a soft object with different

magnitudes for different directions controlled by the dilating

distance function, and it also offers a parameter to control the

degree to which a soft object can be enlarged.

The remainder of this paper is organized as follows.

Section II reviews soft object modeling. Section III

introduces erosion and dilation operations. Conclusion is

given in Section IV.

II. SOFT OBJECT MODELING

This section presents some definitions about soft objects

and implicit surfaces.

A. Field Functions

Let a field function be denoted as fi(x, y, z):R3R+ where

R+ [0, ]. Then, a primitive soft object is represented as the

point set

{(x,y,z)R3| fi(x,y,z)=0.5, i=1,2,…,}

In the following, the symbol fi(x, y, z)=0.5 denotes a soft

object or its surface for short. To enable that fi(x, y, z)=0.5,

i=1, 2,…, can be blended by summation operation, a fi(x, y, z)

is usually defined as (P。di)(x,y,z) and is written by

fi(x,y,z)=(P。di)(x,y,z)= P(di(x,y,z)). (1)

In Eq. (1), P(d), called potential function, must decrease

from 1 to 0 as d increases from 0 to 1 and P(0.5)=0.5, such as

Eq. (2) in [6]:

P(d)=

15.0))45.1(75.0/()1(

5.0))45.4(/()3(1222

222

ddssd

ddssd , (2)

Distance Function Operations for Developing New Field

Functions in Soft Object Modeling

Pi-Chung Hsu

International Journal of Computer and Communication Engineering, Vol. 2, No. 2, March 2013

179

where parameter s controls the softness affect. Fig. 1(a) shows

the shape of p=P(d) of Eq. (2) in d-p plane.

As for di(x, y, z), it is called distance function and is

required to map R3 into [0, ]. In fact, it controls the shape

and size of a soft object fi(x, y, z)=0.5. In addition, di(x, y, z) is

usually defined by using a closed surface di(x, y, z)=1 as

influential region and is calculated by

di(x,y,z)= r / Rd = oIoX / , (3)

where Rd = oI is called the influential radius of (x, y, z),

which is the distance from the origin to the intersecting point

of the vector X=(x, y, z) with the influential region di(x, y,

z)=1 as show= oI n in Fig. 1(b), and r= oX is (x2+y2+z2)0.5.

Subscript d symbolizes that Rd is the influential radius of di(x,

y, z)=1 as the influential region. Due to P(0.5)=0.5, the shape

of a soft object fi(x, y, z)=(P。di)(x, y, z)=0.5 is like the shape

of di(x, y, z) =0.5, and consequently a soft object fi(x, y, z)=0.5

is viewed as the influential region di(x, y, z)=1 scaled overall

by 0.5. Thus, one can say that the size and shape of di(x, y,

z)=1 determines the shape and size of a soft object fi(x, y,

z)=0.5. Some famous distance functions are listed as follows:

0.2 0.4 0.6 0.8 1

d

0.2

0.4

0.6

0.8

1

P

(a)

(b)

Fig. 1. (a). The shape p=P(d) in Eq. (2), which becomes concave downward

and upward gradually as softness parameter s increases. (b). The influential

region di(x,y,z) =1, solid line, and the shape fi(x,y,z)=0.5, i.e. di(x,y,z)=0.5,

dotted line.

Superellipsoids:

d(x,y,z)=(|x/a|p+|y/b|p+|z/c|p)1/p . (4)

Superquadrics [3]:

d(x,y,z)=((|x/a| p1+|y/b| p1) p2/ p1+|z/c| p2)1/p2 . (5)

B. Blending Operations

Blending operations play a very important role in creating

a complex soft object because they can smoothly connect k

primitive soft objects fi(x, y, z)=0.5, i=1,2,…,k via a blending

operator Bk(x1,...,xk):R+k R+ . Precisely, they are written by

{(x, y, z)R3 | Bk(f1(x, y, z),...,fk(x, y, z))=0.5}.

Many blending operations have been developed and they

include:

Soft blending [4, 5]:

Bk(x1,…,xk)=x1+x2+…+xk,

Super-ellipsoidal union [12]:

Bk(x1,…,xk)=(x1p+…+xk

p)1/p ,

Super-ellipsoidal intersection [12]:

Bk(x1,…,xk)=(x1-p +…+xk

-p)-1/p.

Besides, sequential blending operations are also allowed

and they can be represented as a CSG tree [13]. Fig. 2

displays a wheel created from a union of two cylinders and a

toroid.

C. Ray-Linear Functions

As shown in Eq. (3), a distance function is required to be

rewritten by using influential radii. This subsection presents a

theorem to check to see if a function is able to fulfill this

requirement. Now, as stated in [14], the following definition

is introduced first.

Fig. 2. A wheel created from a union of two cylinders and a toroid.

Definition 1: A function f(x, y, z):R3R+ is called

non-negative ray-linear if f(ax, ay, az)=af(x, y, z) holds for

any (x, y, z)R3 and a R+. For compactness, “ray-linear” is

used to stand for “non-negative ray-linear”. Based on

Definition 1, Theorem 1 was proposed in [1] and is listed

below.

Theorem 1: If f(x, y, z):RnR+ is ray-linear, then f(x, y, z)

can be reformulated as r/Rf, where r=||(x, y, z)|| and Rf is the

distance from the origin to the intersecting point of the vector

(x, y, z) with the influential region f(x, y, z)=1.

According to Theorem 1, we obtain that one can adopt

ray-linear f(x, y ,z) as a distance function because it can be

rewritten by r/Rf. Thus, based on Theorem 1, Theorem 2 is

described below for helping check to see if a blending

operation Bk(f1(x, y, z),..., fk(x, y, z)) can be a distance

function.

Theorem 2: If fi(x, y, z):R3R+, i=1,...,k, all are ray- linear

and operator Bk(x1,...,xk):R+kR+ is ray-linear, then operation

Bk(f1(x, y, z),...,fk(x, y, z)) is ray-linear and is a distance

function, too.

International Journal of Computer and Communication Engineering, Vol. 2, No. 2, March 2013

180

III. EROSION AND DILATION OPERATIONS OF DISTANCE

FUNCTIONS

This section explains why and how erosion and dilation

operations can be used to deform a soft object.

A. Erosion Operation

Let d(x, y, z) and T(x, y, z) both be ray-linear distance

functions, and surfaces d(x, y, z)=1 and T(x, y, z)=1 both be

closed and satisfy the condition:

{(x, y, z)R3|d(x, y, z)=1}{(x, y, z)R3|T(x, y, z)=1}.

Thus, an operation of d(x, y, z) an T(x, y, z), denoted as

(dT)(x, y, z) and called erosion operation, is given by

(dT)(x, y, z) = (d(x, y, z)-n - T(x, y, z)-n)1/-n (6)

= Br2(d(x, y, z), T(x, y, z)),

Br2(x1, x2)=(x1-n - x2

-n) 1/-n,

where T(x,y,z) is called eroding function and n1. It is trivial

to show Br2(x1, x2) in Eq. (6) is ray-linear. Since d(x,y,z),

T(x,y,z) and Br2(x1,x2) all are ray-linear, it follows from

Theorem 2 that (dT)(x,y,z) is ray-linear, too and so (d

T)(x,y,z) can be rewritten as (dT)(x,y,z)=r/RdT. This means

that (dT)(x,y,z) in Eq. (6) can be used as a new distance

function to define a new soft object by

(P。(dT)) (x, y, z)=0.5

In fact, (P。(dT)) (x, y, z)=0.5 is viewed as (P。d) (x, y,

z)= 0.5 eroded by the influential radii of T(x, y, z)=0.5. This is

explained as follows. Due to d(x, y, z)=r/Rd and T(x, y,

z)=r/RT from Theorem 1, putting r/Rd and r/RT intoEq. (6)

yields

(dT)(x, y, z)= r / (Rdn -RT

n)1/n, (7)

i.e. (dT)(x, y, z)= d( Rd / (Rdn -RT

n)1/n (x, y, z) ).

From Eq. (7), the shape (dT)(x, y, z)=1 can be viewed as

the shape d(x, y, z)=1 scaled overall with an individual

scaling factor Rd /(Rdn-RT

n)1/n for every point (x, y, z) in all

directions. In fact, scale factor Rd /(Rdn-RT

n)1/n is different for

different (x, y, z) and depends on RT of T(x, y, z)=1 and Rd of

d(x, y ,z)=1. Since the value of Rd /(Rdn-RT

n)1/n is always

greater than 1, operation (dT) (x, y, z) causes reduction in

size on the deformed surface d(X)=1. From Eq. (7), it is also

obtained that influential radius RdT becomes (Rdn-RT

n)1/n after

(dT)(x, y, z). Depending on parameter n, influential radii

RdT are discussed as follows:

In the case that when n=1, (dT) (x, y, z) becomes r/(Rd

-RT), which means the influential radius is reduced to (Rd -RT).

As a result, the surface (dT) (x, y, z)=1 is like the surface

d(x,y,z)=1 where every point (x, y, z) is moved RT inwards to

the origin along the vector [x, y, z] and every (x, y, z) have

different RT. This implies that the surface (dT) (x, y, z)=1 is

like the surface d(x, y, z)=1 eroded (subtracted) by the

influential radii RT of T(x, y, z)=1, which determines the

erosion extent. For example, let d(x, y, z) be (|x/35|2+|y/35|2+

|z/35|2)1/2 and a eroding function T(x, y, z) be a super-

ellipsoid (|x/5|2 +|y/5|2+|z/15|2)1/2. Then, when n=1, the soft

object (P。(d T))(x, y, z)=0.5 in Fig. 3(c) can be viewed as

the sphere (P。d) (x, y, z)=0.5 in Fig. 3(a) eroded by the

object T(x, y, z)= 0.5 in Fig. 3(b).

As n increases from 1 to , influential radius RdT increases

from (Rd -RT) to Rd

. Thus, the erosion effect caused by T(x, y,

z) decreases as n increases. Therefore, the surface (dT)(x, y,

z)=1 dilates from the surface (d(x, y, z)-1-T(x, y, z)-1)-1 =1 to

the surface d(x, y, z)=1 while n increases from 1 to . This is

demonstrated in Fig. 3(d), which follows the example in Fig.

3. Fig. 3(d) shows the soft objects of (P。(dT)) (x, y, z) =0.5

with n set 1, 1.3, 1.7, and 2.5, respectively, for the objects

from left to right.

Consider other cases as shown in Fig. (4) that when d(x, y,

z)=(|x/35|2+|y/35|2+|z/35|2)1/2, T1(x, y, z)=(|x/5|2+|y/15|2+|z

/15|2)1/2 and T2(x, y, z)=((|x/10|7+|y/10|7)2/7+|z/25|2)1/2, then the

erosions of (P。(dT1)) (x, y, z)=0.5 and (P。(dT2)) (x, y,

z)= 0.5 are shown in Figs. 4(d)-(e).

In addition, when the shape of T(x, y, z)=1 is asymmetric,

then the shape of (P。(dT)) (x, y, z)=0.5 can be asymmetric.

As shown in Fig. 5, the asymmetric super-ellipsoid in Fig.

5(b) is used as T(x, y, z)=1 and the sphere in Fig. 5(a) as d(x, y,

z)=1, so the shape of (P。(dT)) (x, y, z)=0.5 is asymmetric

shown in Fig. 5(d). On the contrary, Fig. 5(e) is symmetric in

two of the polar areas because the symmetric super-ellipsoid

in Fig. 5(c) is used as T(x, y, z)=1 instead.

(a) (b) (c)

(d)

Fig. 3. (a). A soft object (P。d) (x, y, z)=0.5 before erosion. (b). An eroding

object T(x, y, z)=1. (c). The erosion of the sphere in (a) by the object in (b)

defined by (P。(dT)) (x, y, z)=0.5. (d). The soft objects of that in (c) with n

in (dT) (x, y, z) is set 1, 1.3, 1.7, and 2.5, respectively, for the objects from

left to right.

(a) (b) (c)

(d) (e)

Fig. 4. The erosions of the sphere in (a) by the disk and the super-ellipsoid in

(b) and (c) generate the two-ball object and the four-ball object respectively

in (d) and (e).

International Journal of Computer and Communication Engineering, Vol. 2, No. 2, March 2013

181

(a) (b) (c)

(d)

(e)

Fig. 5. (a). The shape of d(x, y, z)=0.5: a sphere. (b). The shape of T(x, y,

z)=0.5: an asymmetric super-ellipsoid. (c). The shape of T(x, y, z)=0.5: a

symmetric super-ellipsoid. (d). Soft objects (P。d) (x, y, z)=0.5 with an

asymmetric concave shape because the object in (b) is used as T(x, y, z)=1,

where the left one’s surface is transparent. (e). Soft objects (P。d) (x, y, z)

=0.5 with two symmetric concave polar areas because the object in (c) is used

as T(x, y, z)=1, where the left one’s surface is transparent.

B. Dilation Operation

Let d(x, y, z) and T(x, y, z) be ray-linear distance functions,

and d(x, y, z)=1 and T(x, y, z) =1 both are closed surfaces.

Thus, an operation of d(x, y, z) an T(x, y, z), denoted as (dT)

(x, y, z) and called dilation operation, is given by

(dT)(x, y, z)=(d(x, y, z)-n +T(x, y, z)-n)1/-n (8)

=Br2(d(x, y, z),T(x, y, z))

Br2(x1, x2)=(x1-n + x2

-n) 1/-n

where T(x, y, z) is called dilating function and n1. It is trivial

to show Br2(x1, x2) in Eq. (8) is ray-linear. It follows from

Theorem 1 that (dT) (x, y, z) is also ray-linear since d(x, y,

z), T(x, y z, z) and Br2(x1, x2) all are ray-linear. As a result, (d

T) (x, y, z) can be reformulated as (dT) (x, y, z)=r/RdT. This

means that (dT) (x, y, z) in Eq. (8) can be used as a new

distance function to define a new soft object by

(P。(dT)) (x, y, z)=0.5

Described geometrically, (P 。 (dT)) (x, y, z)=0.5 is

viewed as (P。d) (x, y, z)=0.5 dilated by the influential radii

of the surface T(x, y, z)=0.5. This is proved as follows.

Substituting r/Rd and r/RT for d(x, y, z) and T(x, y, z) in Eq. (8)

yields

(dT)(x, y, z) = r / (Rdn +RT

n)1/n (9)

i.e. (dT)(x, y, z) = d(Rd /(Rdn + RT

n)1/n (x, y, z))

Eq. (9) implies that the shape (dT)(x, y, z)=1 can be

viewed as the shape d(x, y, z)=1 scaled overall with an

individual scaling factor Rd /(Rdn +RT

n)1/n for every point (x, y,

z) in all directions. In fact, scale factor Rd /(Rdn+RT

n)1/n is

different for different (x, y, z) and depends on RT of T(x, y,

z)=1 and Rd of d(x, y, z)=1. Since the value of Rd /(Rdn +RT

n)1/n

is always less than 1, (dT)(x, y, z) =1 causes enlargement in

size of the deformed object d(x, y, z)=1. Eq. (9) indicates that

RdT=(Rdn +RT

n)1/n, that is, the influential radius is raised to

(Rdn+RT

n)1/n after (dT)(x, y, z). Some characteristics of (dT)

(x, y, z) are described by varying the parameter n as follows.

n=1, (dT) (x, y, z) becomes r/(Rd +RT). This

means for any (x, y, z)R3, the influential radius raises to (Rd

+RT). Contrary to (dT)(x, y, z)=1, the surface (dT)(x, y,

z)=1 is like the surface d(x, y, z)=1 where every point (x, y, z)

is moved RT outwards from the origin along the vector [x, y, z]

and every (x, y, z) in different directions have different RT.

This implies that the surface (dT)(x, y, z)=1 is like the

surface d(x, y, z)=1 dilated by the influential radii RT of T(x, y,

z)=1, which determines the dilation extent.

2 n increases from 1 to , the influential radius RdT

decreases from (Rd+RT) to Rd . Consequently, the dilation

effect caused by T(x, y, z) decreases as n increases. Hence, the

surface (dT)(x, y, z)=1 shrinks from the surface (d(x, y, z)-1+

T(x, y, z)-1)-1=1 to the surface d(x, y, z)=1, while n increase

from 1 to . For example, let d(x, y,

z)=(|x/25|2+|y/25|2+|z/25|2)1/2 and T(x, y,

z)=(|x/4|1.1+|y/4|1.1+|z/16|1.1)1/1.1. Thus, the dilation (P 。

(dT))(x, y, z)=0.5 of a sphere by the prism-shaped object in

Fig. 6(b) with n=1, 1.4, 1.8, 2.6 and 5.2, is shown in Fig. 6(c),

where the last object is almost like the original sphere in Fig.

6(a).

(a) (b)

(c)

Fig. 6. The dilation (P。(dT))(x, y, z)=0.5 of the sphere in (a) by the

prism-like object in (b) with n=1, 1.4, 1.8, 2.6, and 5.2, respectively, for the

objects from left to right in (c).

C. Erosion after Dilation Operation

Let d(x, y, z), TE(x, y, z) and TD(x, y, z) be ray-linear

distance functions, then a erosion after dilation operation in

Subsections III A-B above is defined by using TE(x, y, z) and

TD(x, y, z) as a eroding and a dilating functions, and is written

by

(P。((dTD)TE))(x,y,z)=0.5.

Consider the example about the latter one where d(x, y, z)

is (|x/20|2+|y/20|2+|z/20|2)1/2, and TD(x, y, z) is

((|x/5|2+|y/5|2)1.5/2 +|z/15|1.5)1/1.5 to dilate d(x,y,z) around z-axis.

The shapes (P。d)(x, y, z)=0.5 and TD(x,y,z)=0.5 are shown

by the first two object in Fig. 7(a), and the dilation of (P。

(dTD))(x, y, z)= 0.5 is shown in Fig. 7(b), which is larger

than the original one in Fig. 7(a). Thus, to keep unchanged

the region (|x/5|2+ |y/5|2)1/2=0.5 around (P。(dTD))(x, y,

z)=0.5, we can use ((|x/5|2+|y/5 |2)1.5/2+|z/5 |1.5)1/1.5 as TE(x, y, z),

the third object in Fig. 12(a), to restore the size of (P。

International Journal of Computer and Communication Engineering, Vol. 2, No. 2, March 2013

1) When

) As

(dTD))(x, y, z)=0.5 by (P。((dTD) TE))(x, y, z)=0.5. This

is shown in Fig. 7(c) and the resulting shape of (P 。

((dTD)TE))(x, y, z) =0.5 has more similar size around the

region (|x/5|2+|y/5|2)1/2=0.5 to the original soft object (P。

d)(x, y, z)=0.5 in Fig.7(a) , compared to that in Fig. 7(b).

(a)

(b) (c)

Fig. 7. (a) The shapes of (P。d) (x, y, z)=0.5, TE(x, y, z)=0.5 and TD(x, y,

z)=0.5 from left to right, respectively. (b) The dilation (P。(dTD)) (x, y,

z)=0.5. (c) The erosion after dilation (P。((dTD)TE))(x, y, z)=0.5.

IV. CONCLUSION

In soft object modeling, the distance function of a field

function determines the shape and size of primitive soft

objects. To develop distance functions with new shapes,

unlike most of the existing researches focusing directly on

developing new ones with new shapes of iso-surfaces, this

paper has developed a new method that creates a new

distance function by deforming the iso-surface of the distance

function of a soft object via a ray-linear distance function and

hence the soft object is also deformed. More precisely, this

paper has developed the following distance function

operations:

Erosion distance function operation: It is an operation on

the distance function of a soft object intended for

deformation and an eroding distance function. It can erode

the distance function of a soft object with the eroding

distance function, which controls the extent to which every

point in all direction of the soft object can be shrunk.

Dilation distance function operation: It is an operation on

the distance function of a soft object intended for

deformation and a dilating distance function. It can dilate the

distance function of a soft object with the dilating distance

function, which controls the extent to which every point in all

direction of the soft object can be enlarged.

Thus, based on the two operations, one can freely choose

two of any existing point-based distance functions and then

develop a new distance function and a new field function for

a new soft object with a more diverse shape.

ACKNOWLEDGMENT

The author would like to thank National Science Council

of Taiwan for the support of Project No. NSC 101-2221-E-

366-012.

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